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Distributed optimization has received a lot of interest in recent years due to its wide applications in various fields. In this work, we revisit the convergence property of the decentralized gradient descent [A. Nedić-A.Ozdaglar (2009)] on the whole space given bywhere the stepsize is given as α(t) = a (t+w) p with 0 < p ≤ 1. Under the strongly convexity assumption on the total cost function f with local cost functions fi not necessarily being convex, we show that the sequence converges to the optimizer with rate O(t −p ) when the values of a > 0 and w > 0 are suitably chosen.

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In this paper we obtain lower bounds on the radius of spatial analyticity of solutions to the Kawahara equation ut + uux + αuxxx + βuxxxxx = 0, β = 0, given initial data which is analytic with a fixed radius. It is shown that the uniform radius of spatial analyticity of solutions at later time t can decay no faster than 1/|t| as |t| → ∞.